Always state the name of the person with higher rank and present the person of lower rank to them. This can be a bit confusing, but it basically means that you should state the name of the person of higher rank, and then "present" the other person to them. This makes the person of higher rank stand out as the more important person in the situation. Here are some examples:
Introduce a friend or significant other to a relative. The relative has a higher rank: "Dad, I'd like you to meet my boyfriend, Danny."
Introduce a lower-ranking business associate to a higher ranking one: "Mr. CEO, I'd like to introduce Mr. Underling."
Introduce a client to a business associate: "Mr. Client, this is Mr. Money, my associate."
Introduce a younger person to an older one: "Mr. Oldson, I'd like you to meet Sally Youngling."
Introduce a man to a woman: "Mary, this is Jeff."
In a business setting, let rank take precedence over gender. If Mr. Thomas is a higher-ranking male than Mrs. Davis, Mr. Thomas gets the higher authority because of his business position, even though Mrs. Davis is a woman:" Mr. Thomas, may I introduce Mrs. Davis.